Time-dependent Transport Equation Research Articles

Frequency-Dependent Relaxation Time of Electron-Phonon Systems in a Homogeneous Magnetic Field

The problem of electromagnetic-wave absorption in a simple model of metal, consisting of electrons and phonons in the presence of a uniform static magnetic field, is investigated by a quantum-kinetic description. The treatment, which stems from a proper time-dependent transport equation, does not have the time-scale restriction of the usual transition-probability approach and does give a proper description of the time-dependent collective effects in a self-consistent way.

Time-dependent solutions of transport equations

Abstract In transport problems (e.g. conduction in metals) involving a distribution function P, it is usual to assume that when, say, an electric field is applied, a steady state is soon established, and correspondingly we seek solutions of a steady-state transport equation in which we set ∂P/∂t = 0. However, it has already been suggested in the past (e.g. the work of Fröhlich) that dielectric breakdown can arise because electrons may find themselves in an unstable situation, gaining energy from the applied field faster than it can be dissipated through collisions with the lattice. Evidently such possibly unstable situations must depend on the functional behaviour of the relaxation time and the magnitude of the applied field. We felt it would be interesting to see whether, on an elementary basis with a simple and explicitly time-dependent transport equation, unstable [runaway] solutions could be found. This is indeed so and we have determined such solutions for a number of models. We propose a simple criterion for stability (probably too naïve for physical stability) which depends only on the assumption that scattering from any given state is a Markovian process. We notice in particular that there is a limiting form of velocity dependence of the relaxation time which in terms of our model should in itself maintain stability at all applied fields.

Frequency-Dependent Relaxation Time of Electron-Phonon Systems

A derivation of the frequency-dependent relaxation time in a simple model of metal, composed of electrons and phonons, is presented. The problem is investigated by a quantum-kinetic description of the response of the electron-phonon system to an oscillating electric field. The treatment, which stems from a proper time-dependent transport equation, does not have the time scale restriction of the usual transition probability approach, and does give a proper description of the time-dependent collective effects.

Time-of-Flight Investigations of Electron Transport in Gases

Electron swarm experiments may be arranged to satisfy the boundary conditions applicable to a one-dimensional time-dependent transport equation. Under such conditions it may be shown that the distribution of the time-of-arrival of individual electrons at a point on a plane which is L cm from another plane releasing electrons at time t=0 may be interpreted in terms of the diffusion coefficient D and the drift velocity w. The integrals of the distribution functions, i.e., the total numbers of electrons arriving, before and after the addition of an attaching gas, are directly related to the attachment coefficient, β. Since, in such experiments, time is a more accessible independent variable than is length, it is not surprising that much more information on the basic character of electron transport in gases is obtained than can be obtained from a Townsend experiment. Results for values of D, and w as a function of E/P are given for C2H4 and CO2.

Theory of Pulse Propagation in a Laser Amplifier

The growth of a radiation pulse traversing a medium with an inverted population is described by nonlinear, time-dependent photon transport equations, which account for the effect of the radiation on the medium as well as vice versa. The equations are solved in closed form for an arbitrary input pulse and an arbitrary initial distribution of inverted population. The solutions are discussed in detail for the particular cases of a square pulse and a Lorentzian pulse, both with a uniform initial population inversion.

熱中性子化における固有値問題

Eigenvalue problem in time and space dependent thermalization transport equation are summarized. They are important in analysis of time decay constant λ and diiffusion length 1/κ measurements. A study of the Wigner-Wilkins type differential equation for the proton gas moderator indicates the existence of an infinite number of discrete time dependent eigenvalue λκ in the range 0≤λκ<(υΣ)min and of a continuous eigenvalue λ in the range (υΣ)min<λ which has a singular eigenfunction. A method of singular integral equation developed by Muskhelishvili is used in order to prove the existence of the singular eigenfunction in the case of more general scattering kernel. They are compared with a expansion method of a flux in terms of polynomials of energy or of velocity. A boundness of time eigenvalues of infinite and finite medium are illustrated for the both cases of mono energetic neutron and thermalizing neutron. The relevance of Boltzmann transport equation is studied for the very slow neutron. Finally the discussion of the spacial eigenvalue problem is added.

均質核反応系の&lt;i&gt;n&lt;/i&gt;群Boltzmann方程式の解法

核反応で発生した高速中性子が熱中性子に減速される過程を取り扱う方法は多くの著者により考察されてきたが,ここでは減速過程にさらに分裂により高速に戻ってくる中性子の効果を取り入れたn群のBoltzmann方程式を水素を減速材とする均質無限系について正確に解くことを試みた。分裂スペクトルは単色としたが,単色でないときはBoltzmann方程式の線型性から単色のときの解を重畳することによって求めればよい。上述の核反応系での中性子数が時間的に定常の状態に漸近するための条件として,臨界条件が求められ,またそのときの中性子スペクトルが求められる。多群近似を扱うと各グループにてその速度則が v Σa-Σ2/2 則であることが示される。

Combustion Instability: Acoustic Interaction with a Burning Propellant Surface

A theory of the response of a burning solid to a sound wave is developed, based on time-dependent solutions of the transport equations relevant to a combustible having idealized physical and chemical properties. The development is restricted to small perturbations about the steady-state conditions for a rather simple model whose parameters have either a direct physical significance, or are experimentally determinable. The introduction of ad hoc time delays, or other phenomenological parameters has been entirely avoided. Time-dependent reaction rate chemistry also has been avoided, by considering only sufficiently fast reactions that the volumetric rate of reaction corresponds to the steady-state rate appropriate to the instantaneous local conditions as described by solution of the time-dependent transport equations. The predictions of the theory are discussed briefly, and several rather general observations are made. However, in view of the complexity inherent in the phenomenon and the present state of inadequate available empirical knowledge regarding this phenomenon, it is not yet possible to make a quantitative comparison between the theory and experiment.

The Application of Reactor Kinetics to the Analysis of Experiments

The conventional kinetics equations for reactors in which fuel is stationary are derived without approximation from the time-dependent transport equation. The utility of the precise form which results is discussed, first generally, then with reference to the detailed analysis of two typical experiments. It is shown that the analysis required for a precise interpretation of kinetics experiments can generally be made using the same techniques employed in the computation of criticality for stationary cases.

On the formulation and analysis of numerical methods for time dependent transport equations

On the formulation and analysis of numerical methods for time dependent transport equations